Date/Time : 24 Jul 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Shidong Jiang, Flatiron Institute

In this talk, we will first present an overview of fast integral equation methods (FIEMs). Fast algorithms, such as the fast multipole methods and their descendants, have made profound impacts on many areas of scientific computing. Boundary integral equations of the second kind are the natural formulation for boundary value problems of elliptic partial differential equations due to the reduction of dimensionality by one, automatic satisfaction of conditions at infinity for exterior and scattering problems, and well conditioning. The computational bottleneck that arises because the resulting linear system is dense due to the nonlocal and long-range nature of integral operators is largely removed by fast algorithms. FIEMs have been highly successful in solving a large class of problems in fluid mechanics, elasticity, acoustics, and electromagnetics.

Second, we introduce a new class of multilevel, adaptive, dual-space methods for computing fast convolutional transforms. The DMK (dual-space multilevel kernel-splitting) framework uses a hierarchy of grids, computing a smoothed interaction at the coarsest level, followed by a sequence of corrections at finer and finer scales until the problem is entirely local, at which point direct summation is applied. Unlike earlier multilevel summation schemes, DMK exploits the fact that the interaction at each scale is diagonalized by a short Fourier transform, permitting the use of separation of variables without relying on the FFT. The DMK framework substantially simplifies the algorithmic structure of the fast multipole method (FMM) and unifies the FMM, Ewald summation, and multilevel summation, achieving speeds comparable to the FFT in work per grid point, even in a fully adaptive context.

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Date/Time : 12 Jul 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Zhang Zhen, Southern University of Science and Technology

Anisotropic surface diffusion is recently investigated in detail from both modeling and numerical perspectives. While sharp-interface models provide a clear description of the problem, it introduces difficulties in numerical simulations especially when there are topological changes. In this talk, we will present a phase-field approach for modeling anisotropic surface diffusions. We first consider the equilibrium problem of energy minimization. By employing a three-operator splitting technique, we could numerically compute the equilibrium shapes of crystals (ESC) using an optimization approach with mathematical convergence. Then we propose a second-order mass preserving dynamic model as a gradient flow of the anisotropic energy for simulating surface diffusion processes. For surface diffusions in contact with solid substrates, we discuss a regularized thin film model as an approximation of the sharp-interface problem. Numerical results demonstrate the good convergence of these models.

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Date/Time : 29 May 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Renbo Zhao, University of Iowa

We present and analyze a new Frank–Wolfe method for minimizing a -log-homogenous self-concordant barriers, with applications including positron emission tomography, D-optimal design, TV-regularized Poisson image de-blurring, quantum state tomography. The iteration complexity of our method is essentially , which recovers that obtained by Khachiyan (1996) on the D-optimal design problem. In addition, we also present and analyze an away-step variant of our proposed Frank–Wolfe method, and we show the global linear convergence of this method. When specialized to the D-optimal design problem, this settles an open problem in Ahipasaoglu, Sun and Todd (2008).

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Date/Time : 31 Jul 2023 15:00
Venue : S17 #04-05 (SR2)
Speaker : Gao Weiguo, Fudan University

In this talk, we first introduce the shiftedCholeskyQR3 algorithm which has been proposed as an optional choice of the orthogonalization algorithms recently. We extend the error analysis result for oblique inner product and show that the new error bound is optimal. We discuss the loss of orthogonality and validate our conclusions through numerical experiments.

Additionally, we present a mixed precision Jacobi algorithm for computing the singular value decomposition (SVD) of a dense matrix. After appropriate preconditioning, the proposed algorithm computes the SVD in a lower precision as an initial guess, and then performs one-sided Jacobi rotations in the working precision as iterative refinement. By carefully transforming a lower precision solution to a higher precision one, our algorithm achieves about 2x speedup on the x86-64 architecture compared to the usual one-sided Jacobi SVD algorithm in LAPACK, without sacrificing the accuracy.

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Date/Time : 21 Apr 2021 15:00
Venue : via Zoom
Speaker : Andrea Granelli, Credit Suisse

Join Zoom Meeting https://nus-sg.zoom.us/j/82328711543?pwd=WEllYTc2aE1IbU5ETEk3dERKV1JHQT09 Meeting ID: 823 2871 1543 Passcode: 266585 Understanding variance risk is of key importance in mathematical finance since it affects risk management, asset allocation and derivative pricing. Variance risk is priced in financial markets by the so-called variance risk premium (VRP), which refers to the premium demanded for holding assets whose variance is exposed to stochastic shocks. This paper identifies a new modelling framework for equity indices and presents for the first time explicit analytical formulas for their VRP in a multivariate stochastic volatility setting, which includes multivariate non-Gaussian Ornstein-Uhlenbeck processes and Wishart processes. Moreover, we propose to incorporate contagion within the equity index via a multivariate Hawkes process and find that the resulting dynamics of the VRP represent a convincing alternative to the models studied in the literature up to date. We show that our new model can explain the key stylized facts of both equity indices and individual assets and their corresponding VRP, while popular (multivariate) stochastic volatility models fail. We finally prove the existence of a structure-preserving risk neutral measure for our model. In particular, we establish the class of equivalent probabilities that preserve the self-affecting structure of the Hawkes process.

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Date/Time : 14 Apr 2021 16:00
Venue : via Zoom
Speaker : Chaithanya Bandi, National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/86983811912?pwd=NDdLMmlEUE1sSWJ0bnVFc3dNSUJ6UT09&from=addon Meeting ID: 869 8381 1912 Password: 628006 We consider the problem faced by an e-retailer that needs to display a limited set of products to a customer with no prior information. In this context, the e-retailer is allowed to query preferences in order to inform its display. A standard approach to this problem follows a two-step approach: First, estimate the preferences of the customer using a choice model, and then optimize the product display. While this approach is applicable to many settings with stationary customer preferences, this is not applicable to scenarios with changing customer preferences. In this paper, we develop a novel product-driven online framework for efficiently learning customer preferences using a structured questionnaire design. We demonstrate that our approach provably outperforms state-of-the-art methods which focus on eliciting the preference vector. Further, we formulate a robust algorithm for eliciting the optimal display set when the customer responses are noisy.   We establish theoretical foundations for our question-design mechanism and develop efficiency guarantees for our product-driven algorithm. We also present results of our implementation on a real data set obtained from a major fashion retailer.  We demonstrate that we are able to efficiently and customer preferences to inform the optimal product display, and outperform existing approaches based on “estimate, then optimize”. Joint work with Yam Huo (NU), and based on work with Jonathan Amar and Nikos Trichakis (MIT).

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Date/Time : 07 Apr 2021 16:00
Venue : via Zoom
Speaker : Guanghui HU, University of Macau

Join Zoom Meeting: https://nus-sg.zoom.us/j/82872451270?pwd=dVVtVFhacFFIc0p3eWc4TmZ2Yngzdz09&from=addon Meeting ID: 828 7245 1270 Password: 085195 In most cases the ground state would be used as an initial state in a dynamic study of a many-body electronic structure. In the numerical study with density functional theory, a complete simulation would consist of obtaining the ground state of the system by solving Kohn-Sham equation and propagating the system by solving time-dependent Kohn-Sham equation. Although there have been many libraries which can handle two processes in a unified framework, the employment of mesh adaptivity towards the efficiency of the simulation would make the realization of the algorithm much more complicated. In this talk, based on our library AFEABIC, a unified realization of the simulation of above process would be introduced. Several mathematical and technical issues, as well as our methods, would be introduced.

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Date/Time : 31 Mar 2021 16:00
Venue : via Zoom
Speaker : Zhou Zhennan, BICMR, Peking University

Join Zoom Meeting https://nus-sg.zoom.us/j/86252466655?pwd=Z3MxTlJIQTRBQnc1NGs3TXpDOHIzdz09&from=addon Meeting ID: 862 5246 6655 Password: 853012 We investigate the continuum limit that the number of beads goes to infinity in the ring polymer representation of thermal averages. Studying the continuum limit of the trajectory sampling equation sheds light on possible preconditioning techniques for sampling ring polymer configurations with large number of beads. In the case where the potential is quadratic, we show that the continuum limit of the preconditioned mass modified Langevin dynamics converges to its equilibrium exponentially fast, which suggests that the finite-dimensional counterpart has a dimension-independent convergence rate. An efficient sampling method, the pmmLang+RBM, is proposed to compute the quantum thermal average in the interacting quantum particle system. Benefiting from the random batch method (RBM), the pmmLang+RBM reduces the complexity due to the interaction forces per timestep from O(NP^2) to O(NP), where N is the number of beads and P is the number of particles. We also propose an extension of the pmmLang+RBM, which is based on the splitting Monte Carlo method and is applicable when the interacting potential contains a singular part.

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Date/Time : 25 Mar 2021 16:00
Venue : via Zoom
Speaker : Joe Kileel, University of Texas at Austin

Join Zoom Meeting https://nus-sg.zoom.us/j/88213418159?pwd=K2lPSXEvemZGSXhiVGYxTVUzQ0d3QT09 Meeting ID: 882 1341 8159 Password: pcasvd Tensors are higher-order matrices, and decomposing tensors can reveal structure in datasets. In recent years, tensor decomposition has found applications in statistics, computational imaging, signal processing, and quantum chemistry. In this talk, we will present a new numerical method for low-rank symmetric tensor decomposition, building on the usual power method and ideas from classical algebraic geometry. The approach achieves a speed-up over the state-of-the-art by roughly one order of magnitude. We will also describe an “implicit” variant of the algorithm for the case of moment tensors which avoids the explicit formation of higher-order moments, analogously to matrix-free techniques in linear algebra. Time permitting, we will include a quantitative analysis of the non-convex optimization landscape associated with our algorithm. This is based on joint works with Joao Pereira, Tammy Kolda and Timo Klock.

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